# Commutator of the matrices of the rotation group

• spaghetti3451
In summary, the commutator of the matrices of the rotation group is a mathematical operation denoted by [A, B] that represents the difference between two successive rotations. It is important in understanding the algebraic structure of rotations in three-dimensional space and is calculated by multiplying two matrices and subtracting the result from the product in the opposite order. In physics, it plays a crucial role in understanding the behavior of particles and systems under rotations and can be extended to higher dimensions.
spaghetti3451
Consider the rotation group ##SO(3)##.

I know that ##R_{x}(\phi) R_{z}(\theta) - R_{z}(\theta) R_{x} (\phi)## is a commutator?

But can this be called a commutator ##R_{z}(\delta \theta) R_{x}(\delta \phi) R_{z}^{-1}(\delta \theta) R_{x}^{-1} (\delta \phi)##?

I haven't seen anyone use that term for anything other than expressions of the form AB-BA.

I found it in page 31 of Ryder's 'Quantum Field Theory' - second edition.

## 1. What is the commutator of the matrices of the rotation group?

The commutator of the matrices of the rotation group is a mathematical operation that involves multiplying two matrices and then subtracting the result from the product of the two matrices in the opposite order. It is denoted by [A, B] where A and B are matrices. In the context of the rotation group, it represents the difference between two successive rotations.

## 2. Why is the commutator of the matrices of the rotation group important?

The commutator of the matrices of the rotation group is important because it allows us to understand the algebraic structure of rotations in three-dimensional space. It helps us to study the properties and relationships between different rotations and their corresponding matrices.

## 3. How is the commutator of the matrices of the rotation group calculated?

The commutator of the matrices of the rotation group is calculated by multiplying the two matrices and then subtracting the result from the product of the two matrices in the opposite order. This can be represented mathematically as [A, B] = AB - BA.

## 4. What is the significance of the commutator of the matrices of the rotation group in physics?

In physics, the commutator of the matrices of the rotation group is significant because it plays a crucial role in understanding the behavior of particles and systems under rotations. It is used to calculate the angular momentum of a system and to study the symmetries and transformations in physical laws.

## 5. Can the commutator of the matrices of the rotation group be extended to higher dimensions?

Yes, the commutator of the matrices of the rotation group can be extended to higher dimensions. In fact, it is a fundamental concept in the study of rotations in any number of dimensions. The commutator follows the same mathematical rules and properties in higher dimensions as it does in three dimensions.

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